Hardness of approximation, aka inapproximability.

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inapproximability of logarithic factor of indepence set [on hold]

The hardness result derived using PCP theorem for Independent set suggests that there exists some absolute constant $\epsilon_0$ such that for $0< \epsilon < \epsilon_0$, it is hard to ...
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59 views

Is graph coloring complete for poly-APX?

Is the graph coloring problem complete for poly-APX under C-reductions (alternatively, under AP-reductions)? For the graph coloring problem, speaking of a feasible solution means a proper coloring for ...
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45 views

How can I show that zero-one programming is not in APX?

How can I show that zero-one programming is not in APX? Vertex Cover Problem is in the APX class. So can I try a PTAS-reduction from the zero-one programming problem to Vertex Cover and show that ...
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158 views

Approximating #P-hard problems

Consider the classical #P-complete problem #3SAT, i.e., to count the number of valuations to make a 3CNF with $n$ variables satisfiable. I am interested in the additive approximability. Clearly, there ...
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155 views

Complexity of linear programming

It is known that Linear Programming (LP) is P-complete. I am interested in approximation algorithms for LP. There are numerous inapproximability results for NP optimization problems, e.g. it is ...
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120 views

Calculating exact/approximate solution to a formula

Suppose we have a set of variable $\mathbf{y} = \left(y_1, ..., y_n \right)$. Also consider the set of functions $g_i(y_i), 1 \leq i \leq n$. Note that $g_i()$ is dependent only on $y_i$. Consider ...
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91 views

Hardness of approximating chromatic number of triangle-free graphs

The chromatic number of graph, $\chi( G)$ is hard to approximate for general graphs. Are there results of hardness of approximating $\chi(G)$ for triangle-free graphs?
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113 views

Approximate c-chromatic number, each color class is P4-free (cograph)

The classic chromatic number of graph, $\chi(G)$, describes the minimum number of colors needed so that each color class is an independent set. There are many other graph coloring definitions. One of ...
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29 views

Approximation algorithm to a minimization problem with optimization function going to zero

I have a simple function to minimize, but it is not discrete: $f(x) = 2x - \alpha x - c(x)$, where $x$ is a natural number, $\alpha$ is a rational number, $0 \leq \alpha \leq 1$, and $0 \leq c(x) \leq ...
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317 views

Does PSPACE-completeness imply approximation hardness?

It is mentioned in a comment in another cstheorySE post that PSPACE-completeness imply APX-hardness. Can anyone please explain/share a reference for it? Is this "tight"? (i.e., are there ...
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170 views

Planted Clique in G(n,p), varying p

In the planted clique problem, one must recover a $k$-clique planted in an Erdos-Renyi random graph $G(n,p)$. This has mostly been looked at for $p=\frac{1}{2}$, in which case it is known to be ...
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Set packing with maximum coverage objective

We are given a universe $\mathcal{U}=\{e_1,..,e_n\}$ and a set of subsets $\mathcal{S}=\{s_1,s_2,...,s_m\}\subseteq 2^\mathcal{U}$. Set-Packing asks how many disjoint sets we can pack, and is defined ...
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56 views

Bellman principle and approximability

Does anybody know if a combinatorial optimzation problem that enjoys the Bellman's optimality principle can in automatic way be approximated?
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164 views

Comparing graphs

I am looking for a kick in the right direction. What i am trying to do is to compute "similarity" between two graphs, where I define "similarity" as the number of shared paths. example: ...
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1answer
181 views

Almost always almost right

I am looking for a complexity class that relates to APX as BPP relates to P. I have already asked the same question here, but perhaps TCS would be a more fruitful location for answers. The reason for ...
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177 views

Variants of Densest Subgraph Problems

Given a graph $G$, and a vertex-induced subgraph $H$ of $G$, there are two superficially-similar definitions ways to define the density of $H$: (1) Average degree of vertices in $H$ (2) What I will ...
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248 views

Approximation in subexponental time

There are studies about approximation algorithms for NP complete problems in Polynomial time and exact algorithms in exponential time. Are there studies about approximation algorithms for NP complete ...
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Approximating BLEDP on restricted graph classes

In the edge-disjoint paths (EDP) problem, we are given an undirected graph $G$, and a set $\{ (s_i,t_i) \mid 1 \leq i \leq k \}$ of $k$ source-sink pairs. The objective is to maximize the number of ...
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Approximation algorithms for multicut for special classes of graphs

The multicut problem is the following. Given a graph $G=(V,E)$ with edge costs $c_e$ for edge $e$ and a set of $k$ terminal pairs $\{(s_1,t_1),\ldots,(s_k,t_k)\}$, the objective is to find a set of ...
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105 views

The non-metric k-median problem

It is well-known that the non-metric $k$-median problem cannot be approximated better than $O(\log(n))$ (by a gap preserving reduction from the set cover problem). Is there any logarithmic ...
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413 views

Reference Request: Asymptotic hardness of $hk$ coloring $k$-colorable graphs

I heard of a result in approximate graph coloring, but cannot find the source. The result is: For every constant $h$ there exists a sufficiently large $k$ such that coloring a $k$-colorable graph ...
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254 views

Multi prover, verifier games and PCP theorem

This question came up while I was going through Siu On Chan's paper on Approximation Resistance. My question is not really related to the paper though. I also guess that this is more of a reference ...
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199 views

Complexity of finding large grid minors

What is the complexity of finding the largest $k\times k$ grid graph that is a minor of a given graph $G$? It is FPT in $k$, and it seems likely to be NP-hard (or NP-complete in a decision version ...
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763 views

Hardest problems to approximate

Under some assumptions, it is hard to approximate MAX-CLIQUE within a factor $n^{1-\epsilon}$ for any $\epsilon >0$. Are there any other problems that are known to be equally hard to approximate? ...
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number of PCP queries

we know from the PCP theorem that $PCP[O(log(n)),O(1)]=NP$,what if we choose specific number of queries will the theorem hold ?
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285 views

Smoothed analysis of approximation algorithms

Smoothed analysis has been applied many times to understand the runtime of exact algorithms for many problems like linear programming and k-means. There are fairly general results in this realm, for ...
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652 views

How to start studying topics Hardness of approximation and PCP's

Recently I have done an introductory course on complexity theory ( which covered 90% of sipser text book). Now I would like to study the topics Hardness of approximation and PCP's. Can you please ...
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230 views

Hitting sets with a subfamily

Let $F$ be a family of $d$-element subsets of a finite universe $U$ of objects. A family $H$ of $k$-element subsets of $U$, with $1 \le k < d$, is a $(k,d)$-hitting-set of $F$ if for each $V \in F$ ...
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108 views

Approximation for accumulative set cover

Let $S_1,\ldots,S_m\subseteq U$ be subsets of a set $U$ of size $\lvert U\rvert=n$. Over all permutations $\pi$ of the set $\\{1,\ldots,m\\}$, I want to maximize the quantity \begin{equation} ...
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193 views

Does Dinur's proof of PCP Theorem imply a procedure for reconstructing a witness?

In Section 3.2 of On Syntactic versus Computational Views on Approximability by Khanna, et al., the authors state that an adaptation of the results from Proof Verification and Hardness of ...
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665 views

hardness of approximating the chromatic number in graphs with bounded degree

I am looking for hardness results on vertex coloring of graphs with bounded degree. Given a graph $G(V,E)$, we know that for any $\epsilon>0$, it's hard to approximate $\chi(G)$ within a factor ...
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Hardness of approximating |V|+(size of vertex cover)

I know that UGC implies a hardness of 2 for vertex cover, but is there a way to have this hardness on instances where the size of the vertex cover is at least $(1-\epsilon)\frac{|V|}2$? More generally ...
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Inappoximability status of Max One in Three SAT for satisfiable instances

What is the inapproximability status of Max-One-in-Three SAT for satisfiable instances?
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Hardness of approximation without the PCP theorem

An important application of the PCP theorem is that it yields "hardness of approximation" type results. In some relatively simpler cases one can prove such hardness without PCP. Is there, however, any ...
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173 views

What is the best-known inapproximability result for MIN-3CNF-DELETION?

I am really curious what the best-known inapproximability result is for MIN-3CNF-DELETION. To clarify, this is the problem of minimizing the number of unsatisfied clauses in a CNF SAT formula with at ...
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208 views

Approximability of the genus problem

What is currently known about the approximability of the genus problem? A preliminary search tells me that a constant factor approximation is trivial for sufficiently dense graphs, and an ...
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243 views

Is 3SAT problem APX-hard or not?

Could you point me a reference, an answer or it is an open question?
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256 views

What is meant by “if there exists a $\rho$-approximation algorithm with $\rho < 2$, then P = NP”?

For example, for the $k$-center problem we want to prove that a 2-approximation algorithm is optimal. A proof is presented on page 39 (Theorem 2.4) in Williamson and Shmoys, The Design of ...
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What is UG-hardness, and how is it different from NP-hardness based on the unique games conjecture?

There are many inapproximability results which rely on the unique games conjecture. For example, Assuming the unique games conjecture, it is NP-hard to approximate the maximum cut problem within ...
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Algorithms for Interval Coloring with Capacities and Demands

We are given a graph $G = (V,E)$, where each edge $e \in E$ has a capacity $c_e$. There are a set of $k$ requests $R_1,\ldots,R_k$. Request $R_i$ has a demand $d_i$, and has an interval $I_i = ...
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Is the dominating set problem constant-factor-approximable in undirected path graphs?

I am interested in the complexity of the minimum dominating set problem (MDSP) in some specific graph class. A graph is an undirected path graph if it is the vertex-intersection graph of a family of ...
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292 views

Why the reduction from MINIMUM SET COVER to MINIMUM DOMINATING SET means $c \log n$-inapproximability for MINIMUM DOMINATING SET

There is a well-known reduction from MINIMUM SET COVER to MINIMUM DOMINATING SET provided at http://en.wikipedia.org/wiki/Dominating_set#L-reductions (attributed there to Kann 1992, but seen, for ...
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377 views

Approximation results for 3-partition

The 3-partition as defined here is a strongly NP-complete decision problem. Consider one optimization problem of 3-partition where the $m$ subsets each have at most three elements and a sum of not ...
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814 views

What are good approximation algorithms for the subset sum problem so far?

By "good", I mean either the algorithm provides a relatively tight bound or it has a relatively fast running time. Any reference is welcome.
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208 views

Request for references on multicommodity flow-cut results

This is a somewhat subjective question. I am interested in studying the literature on multicommodity flow-cut results, especially the 'positive' results which show that flow is a good approximation to ...
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Inapproximability of multiterminal cut

In the multiterminal cut the input is a graph $G$ and a subset $T$ of its vertices. The task is to remove the minimum number of edges from $G$ such that there is no path connecting any distinct ...
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Hardness of Approximation results for Special Set Packing Problem Wanted

Is there any inapproximability result for the following NP-hard problem, which is a special case of the weighted Set Packing Problem? The general Set Packing Problem would be: Given A Collection of ...
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125 views

Approximation results on scheduling under an uncommon constraint

I am looking for any approximation results for unrelated multiprocessor scheduling problem with precedence constraints (minimizing makespan), i.e. $R|prec|C_{max}$, with the constraint that some tasks ...
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324 views

Approximating and bounding Ramsey numbers

Calculating the diagonal Ramsey numbers R(s,s) is hard. There is a famous quote from Joel Spencer: Erdős asks us to imagine an alien force, vastly more powerful than us, landing on Earth and ...
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Approximating the diameter of a convex set defined by semidefinite constraints

A convex subset $C$ of $\mathbb{R}^{n^2}$ is given as the set of positive semidefinite $n\times n$ matrices whose coefficients fulfill some affine equations. Now, if you want to minimize a linear ...